6. Series

Watch infinite series converge or diverge through their partial sums, and master the convergence tests.

• The Harmonic Series Diverges — The partial sums $S_n$ follow $\ln n + \gamma$ upward — very slowly, but without end.
• The Euler–Mascheroni Constant — The harmonic numbers $H_n$ diverge like $\ln n$, but their gap $\gamma_n = H_n - \ln n$ settles at $\gamma \approx 0.5772$ — the secret is the convergent series $\sum (\frac1n - \ln(1+\frac1n))$.
• The Basel Problem — Stack the terms $\frac{1}{n^2}$ as 3D bars and watch the staircase of partial sums approach the plane at $\frac{\pi^2}{6}$ without ever touching it. Why $\pi$ appears at all is only sketched at the idea level.
• The Euler Product — Multiplying $\zeta(s) = \sum_{n\ge 1} n^{-s}$ by $(1-2^{-s})$ sieves out the even terms, then $(1-3^{-s})$ removes multiples of 3 — sieving leaves only the product over primes $\prod_p (1-p^{-s})^{-1}$. Watch the partial product over the first $k$ primes approach $\zeta(2) = \pi^2/6$.
• Leibniz Series — The partial sums overshoot and undershoot $\frac{\pi}{4}$ in turn, closing in from both sides. The gap between two neighboring partial sums is exactly the error bound.
• Convergence Tests Toolbox — When the ratio $r_k = a_{k+1}/a_k$ settles to a limit $L < 1$, the series converges — watch the ratios dive to $0$ and $\frac{1}{2}$ in two series, and $\frac{1}{k^2+1}$ tuck itself under $\frac{1}{k^2}$ for the comparison test.
• Raabe's Test — When the ratio drifts to 1 and the ratio test surrenders, the finer quantity $R_n = n(a_n/a_{n+1} - 1)$ settles above or below 1 and decides convergence.
• Kummer's Test — Weighting the ratio by a positive auxiliary sequence $(p_n)$, the sign of $K_n = p_n\frac{a_n}{a_{n+1}} - p_{n+1}$ decides convergence. $p_n = 1$ recovers the ratio test, $p_n = n$ recovers Raabe's test.
• Cauchy Condensation Test — Group the terms $a_n$ into blocks $[2^n, 2^{n+1})$: each block sum is sandwiched between condensed terms $2^n a_{2^n}$, so the original and condensed series share their fate.
• Telescoping Series — Split each term into $\frac{1}{n} - \frac{1}{n+1}$ and the neighbors cancel each other, leaving only the very first 1 and a vanishing tail.
• Geometric Series: Filling the Square — Fill a unit square by always taking half of what remains: the gap halves at every step and the total becomes 1.
• Wallis Product — Multiply the rational factors in one at a time and watch the running product swing above and below $\frac{\pi}{2}$, closing in as the swings shrink.
• Stirling's Approximation — On a log scale, $n!$ and the Stirling approximation nearly coincide, and the relative error shrinks like $\frac{1}{12n}$. Where $\pi$ enters is only sketched at the idea level.
• Grandi Series — The partial sums bounce between 1 and 0 forever, so this series has no sum. $\frac{1}{2}$ is only the average of the partial sums (the Cesàro mean), not a sum of the series.
• Dirichlet's Test — The partial sums of $\sin(kx)$ wobble, trapped inside $1/|\sin(x/2)|$, and the $1/n$ weight sinking to $0$ tames them into the convergent series $\sum_{n=1}^{\infty} \frac{\sin(nx)}{n}$ — Dirichlet's test.
• Abel's Test — If \\\sum a_n\ converges and \(b_n)\ is monotone and bounded, then \\\sum a_n b_n\ converges — the companion to Dirichlet's test. The difference: \(b_n)\ need not tend to 0.